Modeling and Solving the Train Timetabling Problem

Abstract

The train timetabling problem aims at determining a periodic timetable for a set of trains that does not violate track capacities and satisfies some operational constraints. In particular, we concentrate on the problem of a single, one-way track linking two major stations, with a number of intermediate stations in between. Each train connects two given stations along the track (possibly different from the two major stations) and may have to stop for a minimum time in some of the intermediate stations. Trains can overtake each other only in correspondence of an intermediate station, and a minimum time interval between two consecutive departures and arrivals of trains in each station is specified. In this paper, we propose a graph theoretic formulation for the problem using a directed multigraph in which nodes correspond to departures/arrivals at a certain station at a given time instant. This formulation is used to derive an integer linear programming model that is relaxed in a Lagrangian way. A novel feature of our model is that the variables in the relaxed constraints are associated only with nodes (as opposed to arcs) of the aforementioned graph. This allows a considerable speed-up in the solution of the relaxation. The relaxation is embedded within a heuristic algorithm which makes extensive use of the dual information associated with the Lagrangian multipliers. We report extensive computational results on real-world instances provided from Ferrovie dello Stato SpA, the Italian railway company, and from Ansaldo Segnalamento Ferroviario SpA.

Fast Approaches to Improve the Robustness of a Railway Timetable

“…In the Caprara et al [3] setting, a train operator applies for allocating its trains on the infrastructure, and specify a profit for the "ideal timetable" they are asking for. Then the infrastructure manager collects all requests from train operators and computes a feasible timetable maximizing the overall profit, i.e., the difference between the profits of the scheduled trains and a cost-penalty function, which takes into account the deviations of the final timetables with respect to the ideal ones (possibly leaving some trains unscheduled).…”

“…Different ILP models based on a graph representation of the problem were presented in [3,4]. In these papers the problem is modeled by means of a directed acyclic multi-graph, in which nodes correspond to arrival and departure events from the stations and arise at some discretized time instants, and arcs correspond to train stops within a station or to train trips.…”

“…Following [3], the aperiodic TTP can be described as follows: given a railway network, described as a set of stations connected by tracks, and an ideal train timetable, find an actual train schedule satisfying all the operational constraints and having a minimum distance from the ideal timetable.…”

“…In particular, we address the aperiodic (non cyclic) TTP version described in [3]. Our approach combines Linear Programming (LP) with Stochastic Programming (SP) and Robust Optimization techniques.…”

An Exact Decomposition Approach for the Real-Time Train Dispatching Problem

“…Con icts between operations are prevented by simple packing constraints. Examples of applications of (TI) to train optimization can be found in [7], [8], [10], [11], [20], [30], [34]: actually the literature is much wider, and we refer to [14], [23] and [31] for extensive surveys. To our knowledge, basically all these works deal with the track allocation problem, which is solved o -line and where the feasible time periods associated with train routes are strongly limited by the tentative timetable.…”